Acceleration of generalized hypergeometric functions through precise remainder asymptotics
Abstract
We express the asymptotics of the remainders of the partial sums sn of the generalized hypergeometric function q+1Fq through an inverse power series zn nl Σk ck/nk, where the exponent l and the asymptotic coefficients ck may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point z=1. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate.